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Part BCSIR NET December 2024property-a-is-simplicity-in-disguise

Property a is simplicity in disguise

We say that a group G has property (A) if every non-trivial homomorphism from G to any group is injective. Which of the following groups has property (A)?

  1. A.The cyclic group of order 6.
  2. B.The symmetric group S5S_{5}.
  3. C.The alternating group A5A_{5}.✓ Correct
  4. D.The dihedral group with ten elements.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests standard counterexample.

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50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Related counterexample: Converse of Lagrange: d | |G| ⇒ subgroup of order d

More on this topic

The chapter behind this: Normal subgroups, quotients and the isomorphism theorems — free to read

From Groups › Normal subgroups, quotients, isomorphism theorems

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